We want to solve limits that have the form nonzero over zero.
Let’s cut to the chase:
Let’s see what is going on with limits of the form .
- The numerator is positive.
- The denominator approaches zero and is positive.
Hence, the expression will become arbitrarily large as approaches . We can see this in the graph of .
We are now ready for our next definition.
If grows arbitrarily large as approaches and is negative, we write and say that the limit of approaches negative infinity as goes to .
Note: Saying ”the limit is equal to infinity” does not mean that the limit exists. It
merely allows us to describe more precisely the behavior of the function near , than
saying ”the limit does not exist”.
Let’s consider a few more examples.
Canceling a factor of from the numerator and denominator means we can more easily check the behavior of this limit. As approaches from the right:
- The numerator is a positivenegative number.
- The denominator is positivenegative and approaching zero.
This means that
Here is our final example.
- The numerator is a negative number.
- The denominator is positive and approaching zero.
Hence our function is approaching from the right.
As approaches from the left,
- The numerator is negative.
- The denominator is negative and approaching zero.
Hence our function is approaching from the left. This means We can confirm our results of the previous two examples by looking at the graph of :
Some people worry that the mathematicians are passing into mysticism when we talk about infinity and negative infinity. However, when we write all we mean is that as approaches , becomes arbitrarily large and becomes arbitrarily large, with taking negative values.